Lesson 6
Welcome
Notes:
Scatterplot Review
library(tidyverse)## ── Attaching packages ──────────────────────────────────────────────────────────── tidyverse 1.2.1 ──## ✔ ggplot2 2.2.1 ✔ purrr 0.2.4
## ✔ tibble 1.4.2 ✔ dplyr 0.7.4
## ✔ tidyr 0.7.2 ✔ stringr 1.2.0
## ✔ readr 1.1.1 ✔ forcats 0.2.0## ── Conflicts ─────────────────────────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()ggplot(data=subset(diamonds,carat<quantile(carat,.99)&price<quantile(price,.99)),aes(x=carat,y=price))+
geom_point(fill=I('#F79420'),color=I('black'),shape=21)+
stat_smooth(method='lm')
Price and Carat Relationship
Response: price and carat are likely positively correlated, but it is not a linear relationship necessarily.
Frances Gerety
Notes:
A diamonds is forever
The Rise of Diamonds
Notes: most successful ad campaign of the 20th century
ggpairs Function
Notes:
# install these if necessary
# install.packages('GGally')
# install.packages('scales')
# install.packages('memisc')
# install.packages('lattice')
# install.packages('MASS')
# install.packages('car')
# install.packages('reshape')
# install.packages('plyr')
# load the ggplot graphics package and the others
library(ggplot2)
library(GGally)##
## Attaching package: 'GGally'## The following object is masked from 'package:dplyr':
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## nasalibrary(scales)##
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## col_factorlibrary(memisc)## Loading required package: lattice## Loading required package: MASS##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
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## select##
## Attaching package: 'memisc'## The following object is masked from 'package:scales':
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## percent## The following objects are masked from 'package:dplyr':
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## as.array# sample 10,000 diamonds from the data set
set.seed(20022012)
diamond_samp <- diamonds[sample(1:length(diamonds$price), 10000), ]
ggpairs(diamond_samp,
lower = list(continuous = wrap("points", shape = I('.'))),
upper = list(combo = wrap("box", outlier.shape = I('.'))))## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
What are some things you notice in the ggpairs output? Response: relationships between price and color and price and clarity. also interesting long tails on price and carat, and bimodal distributions of dimensions.
weight is proportional to volume, by density.
The Demand of Diamonds
Notes:higher demand for less expensive diamonds
library(gridExtra)##
## Attaching package: 'gridExtra'## The following object is masked from 'package:dplyr':
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## combineplot1 <- ggplot(data=diamonds,aes(x=price))+
geom_histogram(bins=100)+
ggtitle('Price')
plot2 <- plot1+
scale_x_log10()+
ggtitle('Price (log10)')
grid.arrange(plot1,plot2)
Connecting Demand and Price Distributions
Notes: there is bimodality of the price, maybe a “rich vs poor” type of buyer.
Scatterplot Transformation
ggplot(data=diamonds,aes(x=carat,y=price))+
geom_point()+
scale_y_log10()
Create a new function to transform the carat variable
cuberoot_trans = function() trans_new('cuberoot',
transform =function(x) x^(1/3),
inverse =function(x) x^3)Use the cuberoot_trans function
ggplot(aes(carat, price), data = diamonds) +
geom_point() +
scale_x_continuous(trans = cuberoot_trans(), limits = c(0.2, 3),
breaks = c(0.2, 0.5, 1, 2, 3)) +
scale_y_continuous(trans = log10_trans(), limits = c(350, 15000),
breaks = c(350, 1000, 5000, 10000, 15000)) +
ggtitle('Price (log10) by Cube-Root of Carat')## Warning: Removed 1683 rows containing missing values (geom_point).
Overplotting Revisited
head(sort(table(diamonds$carat),decreasing=T))##
## 0.3 0.31 1.01 0.7 0.32 1
## 2604 2249 2242 1981 1840 1558head(sort(table(diamonds$price),decreasing=T))##
## 605 802 625 828 776 698
## 132 127 126 125 124 121ggplot(aes(carat, price), data = diamonds) +
geom_point(alpha=1/2,size=3/4,position='jitter') +
scale_x_continuous(trans = cuberoot_trans(), limits = c(0.2, 3),
breaks = c(0.2, 0.5, 1, 2, 3)) +
scale_y_continuous(trans = log10_trans(), limits = c(350, 15000),
breaks = c(350, 1000, 5000, 10000, 15000)) +
ggtitle('Price (log10) by Cube-Root of Carat')## Warning: Removed 1691 rows containing missing values (geom_point).
Other Qualitative Factors
Notes:
Price vs. Carat and Clarity
Alter the code below.
# install and load the RColorBrewer package
library(RColorBrewer)
ggplot(aes(x = carat, y = price,color=clarity), data = diamonds) +
geom_point(alpha = 0.5, size = 1, position = 'jitter') +
scale_color_brewer(type = 'div',
guide = guide_legend(title = 'Clarity', reverse = T,
override.aes = list(alpha = 1, size = 2))) +
scale_x_continuous(trans = cuberoot_trans(), limits = c(0.2, 3),
breaks = c(0.2, 0.5, 1, 2, 3)) +
scale_y_continuous(trans = log10_trans(), limits = c(350, 15000),
breaks = c(350, 1000, 5000, 10000, 15000)) +
ggtitle('Price (log10) by Cube-Root of Carat and Clarity')## Warning: Removed 1693 rows containing missing values (geom_point).
Clarity and Price
Response: clarity explains a lot of the variance as there is statification across the band due to clarity. holding carat constant, clarity shows a relationship with price.
Price vs. Carat and Cut
Alter the code below.
ggplot(aes(x = carat, y = price, color = cut), data = diamonds) +
geom_point(alpha = 0.5, size = 1, position = 'jitter') +
scale_color_brewer(type = 'div',
guide = guide_legend(title = 'Cut', reverse = T,
override.aes = list(alpha = 1, size = 2))) +
scale_x_continuous(trans = cuberoot_trans(), limits = c(0.2, 3),
breaks = c(0.2, 0.5, 1, 2, 3)) +
scale_y_continuous(trans = log10_trans(), limits = c(350, 15000),
breaks = c(350, 1000, 5000, 10000, 15000)) +
ggtitle('Price (log10) by Cube-Root of Carat and Cut')## Warning: Removed 1696 rows containing missing values (geom_point).
Cut and Price
Response: cut seems to have less of a relationship to price than clarity did. Most are ideal cut anyway.
Price vs. Carat and Color
Alter the code below.
ggplot(aes(x = carat, y = price, color = color), data = diamonds) +
geom_point(alpha = 0.5, size = 1, position = 'jitter') +
scale_color_brewer(type = 'div',
guide = guide_legend(title = 'Color',
override.aes = list(alpha = 1, size = 2))) +
scale_x_continuous(trans = cuberoot_trans(), limits = c(0.2, 3),
breaks = c(0.2, 0.5, 1, 2, 3)) +
scale_y_continuous(trans = log10_trans(), limits = c(350, 15000),
breaks = c(350, 1000, 5000, 10000, 15000)) +
ggtitle('Price (log10) by Cube-Root of Carat and Color')## Warning: Removed 1688 rows containing missing values (geom_point).
Color and Price
Response: color shows a relationship with price
Linear Models in R
Notes: We apply the log transformation to the price, and the cube root to the carat. Response: log(price)~carat^(1/3)
Building the Linear Model
Notes:
m1 <- lm(I(log(price)) ~ I(carat^(1/3)), data = diamonds)
m2 <- update(m1, ~ . + carat)
m3 <- update(m2, ~ . + cut)
m4 <- update(m3, ~ . + color)
m5 <- update(m4, ~ . + clarity)
mtable(m1, m2, m3, m4, m5)##
## Calls:
## m1: lm(formula = I(log(price)) ~ I(carat^(1/3)), data = diamonds)
## m2: lm(formula = I(log(price)) ~ I(carat^(1/3)) + carat, data = diamonds)
## m3: lm(formula = I(log(price)) ~ I(carat^(1/3)) + carat + cut, data = diamonds)
## m4: lm(formula = I(log(price)) ~ I(carat^(1/3)) + carat + cut + color,
## data = diamonds)
## m5: lm(formula = I(log(price)) ~ I(carat^(1/3)) + carat + cut + color +
## clarity, data = diamonds)
##
## ============================================================================================
## m1 m2 m3 m4 m5
## --------------------------------------------------------------------------------------------
## (Intercept) 2.821*** 1.039*** 0.874*** 0.932*** 0.415***
## (0.006) (0.019) (0.019) (0.017) (0.010)
## I(carat^(1/3)) 5.558*** 8.568*** 8.703*** 8.438*** 9.144***
## (0.007) (0.032) (0.031) (0.028) (0.016)
## carat -1.137*** -1.163*** -0.992*** -1.093***
## (0.012) (0.011) (0.010) (0.006)
## cut: .L 0.224*** 0.224*** 0.120***
## (0.004) (0.004) (0.002)
## cut: .Q -0.062*** -0.062*** -0.031***
## (0.004) (0.003) (0.002)
## cut: .C 0.051*** 0.052*** 0.014***
## (0.003) (0.003) (0.002)
## cut: ^4 0.018*** 0.018*** -0.002
## (0.003) (0.002) (0.001)
## color: .L -0.373*** -0.441***
## (0.003) (0.002)
## color: .Q -0.129*** -0.093***
## (0.003) (0.002)
## color: .C 0.001 -0.013***
## (0.003) (0.002)
## color: ^4 0.029*** 0.012***
## (0.003) (0.002)
## color: ^5 -0.016*** -0.003*
## (0.003) (0.001)
## color: ^6 -0.023*** 0.001
## (0.002) (0.001)
## clarity: .L 0.907***
## (0.003)
## clarity: .Q -0.240***
## (0.003)
## clarity: .C 0.131***
## (0.003)
## clarity: ^4 -0.063***
## (0.002)
## clarity: ^5 0.026***
## (0.002)
## clarity: ^6 -0.002
## (0.002)
## clarity: ^7 0.032***
## (0.001)
## --------------------------------------------------------------------------------------------
## R-squared 0.924 0.935 0.939 0.951 0.984
## adj. R-squared 0.924 0.935 0.939 0.951 0.984
## sigma 0.280 0.259 0.250 0.224 0.129
## F 652012.063 387489.366 138654.523 87959.467 173791.084
## p 0.000 0.000 0.000 0.000 0.000
## Log-likelihood -7962.499 -3631.319 -1837.416 4235.240 34091.272
## Deviance 4242.831 3613.360 3380.837 2699.212 892.214
## AIC 15930.999 7270.637 3690.832 -8442.481 -68140.544
## BIC 15957.685 7306.220 3761.997 -8317.942 -67953.736
## N 53940 53940 53940 53940 53940
## ============================================================================================Notice how adding cut to our model does not help explain much of the variance in the price of diamonds. This fits with out exploration earlier.
Model Problems
Video Notes:
Research: (Take some time to come up with 2-4 problems for the model) (You should 10-20 min on this)
Response: There are a lot of factors which means it could potentially overfit the data. Also, this may not generalize to new data as diamond price likely fluctuates over time due to inflation and demand etc.
this data is from 2008. inflation and the 2008 recession greatly affected diamond prices from that time. china is also now a factor in diamond prices. ***
A Bigger, Better Data Set
Notes:
# install.package('bitops')
# install.packages('RCurl')
# library('bitops')
# library('RCurl')
#
# diamondsurl = getBinaryURL("https://raw.github.com/solomonm/diamonds-data/master/BigDiamonds.Rda")
# load(rawConnection(diamondsurl))The code used to obtain the data is available here: https://github.com/solomonm/diamonds-data
Building a Model Using the Big Diamonds Data Set
Notes:
diamondsBigSample <- read.csv('diamondsbig.csv')
diamondsBigSample<-subset(diamondsBigSample,price<10000 & cert=='GIA')
m1 <- lm(I(log(price)) ~ I(carat^(1/3)), data = diamondsBigSample)
m2 <- update(m1, ~ . + carat)
m3 <- update(m2, ~ . + cut)
m4 <- update(m3, ~ . + color)
m5 <- update(m4, ~ . + clarity)
mtable(m1, m2, m3, m4, m5)##
## Calls:
## m1: lm(formula = I(log(price)) ~ I(carat^(1/3)), data = diamondsBigSample)
## m2: lm(formula = I(log(price)) ~ I(carat^(1/3)) + carat, data = diamondsBigSample)
## m3: lm(formula = I(log(price)) ~ I(carat^(1/3)) + carat + cut, data = diamondsBigSample)
## m4: lm(formula = I(log(price)) ~ I(carat^(1/3)) + carat + cut + color,
## data = diamondsBigSample)
## m5: lm(formula = I(log(price)) ~ I(carat^(1/3)) + carat + cut + color +
## clarity, data = diamondsBigSample)
##
## ================================================================================================
## m1 m2 m3 m4 m5
## ------------------------------------------------------------------------------------------------
## (Intercept) 2.671*** 1.333*** 0.949*** 1.341*** 0.665***
## (0.003) (0.012) (0.012) (0.010) (0.007)
## I(carat^(1/3)) 5.839*** 8.243*** 8.633*** 8.110*** 8.320***
## (0.004) (0.022) (0.021) (0.017) (0.012)
## carat -1.061*** -1.223*** -0.782*** -0.763***
## (0.009) (0.009) (0.007) (0.005)
## cut: Ideal 0.211*** 0.181*** 0.131***
## (0.002) (0.001) (0.001)
## cut: V.Good 0.120*** 0.090*** 0.071***
## (0.002) (0.001) (0.001)
## color: E/D -0.083*** -0.071***
## (0.001) (0.001)
## color: F/D -0.125*** -0.105***
## (0.001) (0.001)
## color: G/D -0.178*** -0.162***
## (0.001) (0.001)
## color: H/D -0.243*** -0.225***
## (0.002) (0.001)
## color: I/D -0.361*** -0.358***
## (0.002) (0.001)
## color: J/D -0.500*** -0.509***
## (0.002) (0.001)
## color: K/D -0.689*** -0.710***
## (0.002) (0.002)
## color: L/D -0.812*** -0.827***
## (0.003) (0.002)
## clarity: I2 -0.301***
## (0.006)
## clarity: IF 0.751***
## (0.002)
## clarity: SI1 0.426***
## (0.002)
## clarity: SI2 0.306***
## (0.002)
## clarity: VS1 0.590***
## (0.002)
## clarity: VS2 0.534***
## (0.002)
## clarity: VVS1 0.693***
## (0.002)
## clarity: VVS2 0.633***
## (0.002)
## ------------------------------------------------------------------------------------------------
## R-squared 0.888 0.892 0.899 0.937 0.969
## adj. R-squared 0.888 0.892 0.899 0.937 0.969
## sigma 0.289 0.284 0.275 0.216 0.154
## F 2700903.714 1406538.330 754405.425 423311.488 521161.443
## p 0.000 0.000 0.000 0.000 0.000
## Log-likelihood -60137.791 -53996.269 -43339.818 37830.414 154124.270
## Deviance 28298.689 27291.534 25628.285 15874.910 7992.720
## AIC 120281.582 108000.539 86691.636 -75632.827 -308204.540
## BIC 120313.783 108043.473 86756.037 -75482.557 -307968.400
## N 338946 338946 338946 338946 338946
## ================================================================================================Predictions
Example Diamond from BlueNile: Round 1.00 Very Good I VS1 $5,601
#Be sure you’ve loaded the library memisc and have m5 saved as an object in your workspace.
thisDiamond = data.frame(carat = 1.00, cut = "V.Good",
color = "I", clarity="VS1")
modelEstimate = predict(m5, newdata = thisDiamond,
interval="prediction", level = .95)
#Need to exp the model estimate because the model is nat log base.
exp(modelEstimate)## fit lwr upr
## 1 5040.436 3730.34 6810.638thisDiamond## carat cut color clarity
## 1 1 V.Good I VS1Evaluate how well the model predicts the BlueNile diamond’s price. Think about the fitted point estimate as well as the 95% CI.
this diamond is decently well fit by the model (5601 actual to 5040 modeled), but the model does have pretty large confidence intervals.
dat = data.frame(m4$model, m4$residuals)
with(dat, sd(m4.residuals))## [1] 0.2164168with(subset(dat, carat > .9 & carat < 1.1), sd(m4.residuals))## [1] 0.2178147dat$resid <- as.numeric(dat$m4.residuals)
ggplot(aes(y = resid, x = round(carat, 2)), data = dat) +
geom_line(stat = "summary", fun.y = sd)## Warning: Removed 1 rows containing missing values (geom_path).
This shows that the model explains aroun 80ish percent of the variance, and has high variance for the larger diamonds.
Final Thoughts
Notes:
Click KnitHTML to see all of your hard work and to have an html page of this lesson, your answers, and your notes!